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and out of relatively immobile water resident in micropores. The mobile-immobile
model involves two coupled equations (in one-dimensional form):
;
o
ð
h
im
c
im
Þ
ot
þ
o
ð
h
m
c
m
Þ
ot
¼
o
ox
ð
vh
m
c
m
Þþ
o
D
h
m
o
c
m
ox
ox
ð
10
:
7
Þ
o
ð
h
im
c
im
Þ
ot
¼ e
ð
c
m
c
im
Þ
where c
m
and c
im
, and h
m
and h
im
denote the mobile and immobile zone con-
centrations and volumetric moisture contents, respectively, and e is a mass-transfer
coefficient. The mobile-immobile approach, while applied frequently, is limited
by its reliance on the Fickian-based advection-dispersion equation. We consider
alternatives to this approach in
Sect. 10.3
.
As an alternative to continuum-based treatments, and to capture the pore-scale
details of such transport, a wide variety of network modeling approaches can be
dispersion phenomena in porous media using network models and percolation
theory is presented by Sahimi (
1987
) and Sahimi and Imdakm (
1988
), who derive a
set of relationships relevant to the geometrical and hydraulic characteristics of
capillary tube networks and use them to establish various expressions for describing
dispersion of contaminants under various conditions and assumptions. For example,
the relationship between mean square contaminant displacement and time yields
different exponents, depending on the flow conditions (e.g., as characterized by the
Peclet number, Pe : vl/D, where v is mean water velocity, l is the characteristic
length—roughly, the mean pore size—and D is the diffusion coefficient; this
dimensionless number gives the ratio of advective to diffusive forces). In particular,
such random networks can be used to examine advective transport through the main
conducting path, with diffusive transport into dead ends and stagnant regions, similar
to the mobile-immobile conceptualization of transport (Koplik et al.
1988
).
We discuss network models further in
Sect. 11.4
, in the context of immiscible
displacements.
10.3 Non-Fickian Transport
The previous sections discussed the advection-dispersion equation and variants
such as the mobile-immobile conceptualization, which are based on the key
assumption that mechanical dispersion is Fickian. In other words, the advection-
dispersion equation (Eq.
10.5
) is strictly valid only under perfectly homogeneous
flow conditions. This requires that the length scale of all heterogeneities be much
smaller than the length scale of the domain. In terms of real transport problems, the
contaminant migration time must be sufficiently large before the advection-
dispersion equation correctly applies. In laboratory-scale columns, homogeneous
